The issue is a little bit complex.
John Etchemendy, TARSKI ON TRUTH AND LOGICAL CONSEQUENCE, in The Journal of Symbolic Logic, Vol.53 (1988)
Mario Gomez-Torrente, Tarski on Logical Consequence, in ND Journal of Formal Logic, Vol.37 (1996)
Craig Bach, TARSKI'S 1936 ACCOUNT OF LOGICAL CONSEQUENCE, in Modern Logic, Vol.7 (1997)
the new translation of Alfred Tarski, On the Concept of Following Logically (1936), in HISTORY AND PHILOSOPHY OF LOGIC, 23 (2002).
I'll try to summarize the central issue, trying to avoid technical details (due also to the lack of LaTeX editor).
Reference is, of course, to Wilfrid Hodges' entry in SEP : Tarski's Truth Definitions.
We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted [...]. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956 [...].
In 1933 Tarski assumed that the formal languages that he was dealing with had two kinds of symbol, namely constants and variables. The constants included logical constants, but also any other terms of fixed meaning. The variables had no independent meaning and were simply part of the apparatus of quantification.
Today logicians work with formal languages, whose extra-logical constants are uninterpreted (until an interpretation is fixed in a particular case). To provide an interpretation or model of such a language, one typically specifies a domain or universe of discourse and assigns to each individual constant a unique object in the domain, to each monadic first-order predicate a subset of the domain, and so on.
Tarski  worked instead with what he called formalized languages, in which the extra-logical constants are interpreted and the domain is fixed.
As Hodges explains :
Model theory by contrast works with three levels of symbol. There are the logical constants (=, ¬, & for example), the variables (as before), and between these a middle group of symbols which have no fixed meaning but get a meaning through being applied to a particular structure. The symbols of this middle group include the nonlogical constants of the language, such as relation symbols [like ∈ in set theory], function symbols and constant individual symbols [like 0 in arithmetic]. They also include the quantifier symbols ∀ and ∃, since we need to refer to the structure to see what set they range over. This type of three-level language corresponds to mathematical usage; for example we write the addition operation of an abelian group as +, and this symbol stands for different functions in different groups.
By the late 1940s it had become clear that a direct model-theoretic truth definition was needed. The version we use today is based on that published by Tarski and Robert Vaught in 1956.
The right way to think of the model-theoretic definition is that we have sentences whose truth value varies according to the situation where they are used. So the nonlogical constants are not variables; they are definite descriptions whose reference depends on the context. Likewise the quantifiers have this indexical feature, that the domain over which they range depends on the context of use.
Because Tarski  uses a formalized language with interpreted extra-logical constants, he must first replace extralogical constants by variables of the same type and then consider what sets of objects satisfy the resulting sentential function.
The concept of a model in Tarski  is not the contemporary concept. In contemporary mathematics, as Hodges points out, a model or structure is roughly a collection of elements with relations defined on them. A set of objects is not such a structure.